Numerical properties of 30

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Show numerical properties of 30

We start by listing out divisors for 30

Divisor	Divisor Math
1	30 ÷ 1 = 30
2	30 ÷ 2 = 15
3	30 ÷ 3 = 10
5	30 ÷ 5 = 6
6	30 ÷ 6 = 5
10	30 ÷ 10 = 3
15	30 ÷ 15 = 2

Positive or Negative Number Test:

Positive Numbers > 0

Since 30 ≥ 0 and it is an integer
30 is a positive number

Whole Number Test:

Positive numbers including 0
with no decimal or fractions

Since 30 ≥ 0 and it is an integer
30 is a whole number

Prime or Composite Test:

Since 30 has divisors other than 1 and itself
it is a composite number

Perfect/Deficient/Abundant Test:

Calculate divisor sum D
If D = N, then it's perfect
If D > N, then it's abundant
If D < N, then it's deficient
Divisor Sum = 1 + 2 + 3 + 5 + 6 + 10 + 15
Divisor Sum = 42
Since our divisor sum of 42 > 30
30 is an abundant number!

Odd or Even Test (Parity Function):

A number is even if it is divisible by 2
If not divisible by 2, it is odd

15  =	30
	2

Since 15 is an integer, 30 is divisible by 2
it is an even number
This can be written as A(30) = Even

Evil or Odious Test:

Get binary expansion
If binary has even amount 1's, then it's evil
If binary has odd amount 1's, then it's odious
30 to binary = 11110
There are 4 1's, 30 is an evil number

Triangular Test:

Can you stack numbers in a pyramid?
Each row above has one item less than the row before it
Using a bottom row of 8 items, we cannot form a pyramid
30 is not triangular

Triangular number: 1st  2nd  3rd  4th  5th  6th  7th  8th  9th  10th  

Rectangular Test:

Is there an integer m such that n = m(m + 1)
The integer m = 5 satisifes our rectangular number property.
5(5 + 1) = 30

Rectangular number: 1st  2nd  3rd  4th  5th  6th  7th  8th  9th  10th  

Automorphic (Curious) Test:

Does n² ends with n
30² = 30 x 30 = 900
Since 900 does not end with 30
it is not automorphic (curious)

Automorphic number: 1st  2nd  3rd  4th  5th  6th  7th  8th  9th  10th  

Undulating Test:

Do the digits of n alternate in the form abab
Since 30 < 100
We only perform the test on numbers > 99

Square Test:

Is there a number m such that m² = n
5² = 25 and 6² = 36 which do not equal 30
Therefore, 30 is not a square

Cube Test:

Is there a number m such that m³ = n
3³ = 27 and 4³ = 64 ≠ 30
Therefore, 30 is not a cube

Palindrome Test:

Is the number read backwards equal to the number?
The number read backwards is 03
Since 30 <> 03
it is not a palindrome

Palindromic Prime Test:

Is it both prime and a palindrome
From above, since 30 is not both prime and a palindrome
it is NOT a palindromic prime

Repunit Test:

A number is repunit if every digit is equal to 1
Since there is at least one digit in 30 ≠ 1
then it is NOT repunit

Apocalyptic Power Test:

Does 2ⁿ contains the consecutive digits 666.
2³⁰ = 1073741824
Since 2³⁰ does not have 666
30 is NOT an apocalyptic power

Pentagonal Test:

It satisfies the form:

n(3n - 1)
2

Check values of 4 and 5

Using n = 5, we have:

5(3(5 - 1)
2

5(15 - 1)
2

5(14)
2

70
2

35 ← Since this does not equal 30
this is NOT a pentagonal number

Using n = 4, we have:

4(3(4 - 1)
2

4(12 - 1)
2

4(11)
2

44
2

22 ← Since this does not equal 30
this is NOT a pentagonal number

Pentagonal number: 1st  2nd  3rd  4th  5th  6th  7th  8th  9th  10th  

Hexagonal Test:

Is there an integer m such that n = m(2m - 1)
No integer m exists such that m(2m - 1) = 30
Therefore 30 is not hexagonal

Hexagonal number: 1st  2nd  3rd  4th  5th  6th  7th  8th  9th  10th  

Heptagonal Test:

Is there an integer m such that:

m  =	n(5n - 3)
	2

No integer m exists such that m(5m - 3)/2 = 30
Therefore 30 is not heptagonal

Heptagonal number: 1st  2nd  3rd  4th  5th  6th  7th  8th  9th  10th  

Octagonal Test:

Is there an integer m such that n = m(3m - 3)
No integer m exists such that m(3m - 2) = 30
Therefore 30 is not octagonal

Octagonal number: 1st  2nd  3rd  4th  5th  6th  7th  8th  9th  10th  

Nonagonal Test:

Is there an integer m such that:

m  =	n(7n - 5)
	2

No integer m exists such that m(7m - 5)/2 = 30
Therefore 30 is not nonagonal

Nonagonal number: 1st  2nd  3rd  4th  5th  6th  7th  8th  9th  10th  

Tetrahedral (Pyramidal) Test:

Satisfies the form:

n(n + 1)(n + 2)
6

Check values of 4 and 5

Using n = 5, we have:

5(5 + 1)(5 + 2)
6

5(6)(7)
6

210
6

35 ← Since this does not equal 30
This is NOT a tetrahedral (Pyramidal) number

Using n = 4, we have:

4(4 + 1)(4 + 2)
6

4(5)(6)
6

120
6

20 ← Since this does not equal 30
This is NOT a tetrahedral (Pyramidal) number

Narcissistic (Plus Perfect) Test:

Is equal to the square sum of it's m-th powers of its digits
30 is a 2 digit number, so m = 2
Square sum of digits^m = 3² + 0²
Square sum of digits^m = 9 + 0
Square sum of digits^m = 9
Since 9 <> 30
30 is NOT narcissistic (plus perfect)

Catalan Test:

Cn  =	2n!
	(n + 1)!n!

Check values of 4 and 5

Using n = 5, we have:

C5  =	(2 x 5)!
	5!(5 + 1)!

Using our factorial lesson

C5  =	10!
	5!6!

C5  =	3628800
	(120)(720)

C5  =	3628800
	86400

C₅ = 42
Since this does not equal 30
This is NOT a Catalan number

Using n = 4, we have:

C4  =	(2 x 4)!
	4!(4 + 1)!

Using our factorial lesson

C4  =	8!
	4!5!

C4  =	40320
	(24)(120)

C4  =	40320
	2880

C₄ = 14
Since this does not equal 30
This is NOT a Catalan number

Property Summary for the number 30

  ·  Positive
  ·  Whole
  ·  Composite
  ·  Abundant
  ·  Even
  ·  Evil
  ·  Rectangular

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Tags:

• divisor
• even
• narcissistic numbers
• number property
• number
• odd
• palindrome
• pentagon
• pentagonal number
• perfect number
• property

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